I watched Thursday's strike in Birmingham, where a rally in Victoria Square, a march and pickets added to a successful day of protest, forcing the national news to shine a light on civil service pensions. Naturally the Times front page claimed "First strikes fail to spark"; the Sun said "Pension walkout a flop"; and the Mail called it "Day Britain defied the militants". But was this the "flop" No 10 claimed, or was it an outrageous public inconvenience as the Mail reported: "11,000 schools hit: The biggest teachers walk-out in a generation". Which way should they play it?

The day's success was not in numbers – always unreliable on all sides – but in exposing the government on the "gold-plated" public sector. Because the coalition does broad-brush bombast, not forensics, Francis Maude walked slap-bang into the same simple error on Today that flummoxed his colleague Justine Greening the day before. He had plainly not read John Hutton's report, perhaps no surprise, with cuts already fixed at £2.8bn before he reported.

As ever, Channel 4's FactCheck lifts the lid: David Cameron was wrong to say public pensions were "going broke"; Maude was wrong to claim costs were rising when Hutton's graph shows costs already falling. The Office for Budget Responsibility and the National Audit Office say there will be no rise. Why? The public accounts committee credits Labour's 2008 raising of civil servants' retirement age from 60 to 65, with many public pensions already switched from final salary to career average – cheaper and fairer to the lower paid. All this was done with negotiation. What's more, Hutton says there is no rush: this can wait until growth is healthier.

That leaves only Cameron's worst argument: public pensions are unfair because they are better than the private sector, where two thirds of employees have none at all. But that has become the lightbulb moment, when people suddenly realise just how many employers contribute nothing, while the taxpayer gives generous relief to the richest: FTSE 100 directors get an average £3.4m pension. Talking to strikers on Thursday, many only recently understood how much will be taken from their pay packets. Jason, a DWP benefits processor earning £18,500, has £45 a month taken in extra contributions: he may drop out. Michael, a careworker on £17,000, loses a similar sum; Doug, a teacher on £29,000, loses £130 a month; and Ashley, a crown prosecution admin officer on £19,000, loses £50 a month. Every one of these, mostly on middling-to-below median pay, face a hefty pay cut during a second year of pay freeze, with inflation at 4 to 5%.

I started out reporting on many industrial disputes through the 1970s, observing bitter conflicts between sometimes brutish managements and sometimes bullying trade unionists. It would have been quite unthinkable to take sums such as these out of people's pay, let alone to be cutting the pensions of those already retired. Ask human resources managers and you would find few companies ready to impose such conditions now, let alone provoke anger the way Danny Alexander did by announcing this is final, mid-negotiation. Surprisingly, the unions are less intransigent than you might expect: most accept there will be further cuts.

Thursday's strike was a reminder of how pitifully weak unions have become, undisguised by fighting words from Mark Serwotka or Dave Prentis promising a reprise of the General Strike. My colleague Aditya Chakrabortty wrote this week of IMF research showing how weak unionism fuels an extreme inequality that endangers economies. Money is sucked upwards, people borrow to survive and credit bubbles erupt. The IMF, hardly on the left, links inequality and weak bargaining for employees with economic instability.

In the decade to 2008, the high pay commission shows how the 34% GDP growth was so unfairly shared that 95% of the people received less while most went to the top 1%. The bottom 10% had nearly ten times less than GDP growth. This year there were 50 times fewer strike days than in the dying days of the 1970s – not 50 days, but 50 times fewer. However, to measure union power by strike days would be a mistake: where unions are strongest and most successful, they don't need to strike. In Germany and Scandinavia they are so woven into management and the national structure that co-operation, not confrontation, benefits the economy. The 1970s ended in catastrophe for trade unionism. Now Labour needs to frame a new plan to give employees constructive power: Ed Miliband wants employees on remuneration committees, but they should be represented in boardrooms too.

This week the Tories tried to resurrect fears of the bad old 1970s – but it didn't work. Cameron tried to paint Miliband as the creature of the unions that elected him: he sidestepped that trap and rightly castigated the government's behaviour over the pensions issue. A bit of history may help: as far as I can discover, no Labour party has ever officially supported a strike, not the General Strike, nor any miners' strike. Shirley Williams was pilloried for joining the Grunwick picket line which later turned violent, but it wasn't Labour policy. Neil Kinnock was tormented for not backing the miners against Margaret Thatcher in 1984, or the six-month-long ambulance strike in 1989-90.

Public sympathy usually wanes: for all the Brassed Off popular romance, Arthur Scargill started out 2:1 against him and ended with 5:1 against. For Labour, a party aspiring to govern can't stand against an elected government, nor easily back producers against the people. Founded and financed by the unions, Labour has always stood apart, but of course that makes the party writhe.

On Thursday, according to Peter Kellner of YouGov, the people swung to support the public workers against Cameron by 50:40. But they didn't support the strike, with 50:40 against. The day of protest made its point forcefully, but unions need to nurture that public backing. Miliband says strikes are a sign of failure, but he needs to map out a better bargaining power for fairer long-term distribution of wealth. As for Cameron, his divide-and-rule strategy will fail. Had he talked to strikers and bystanders this week he would know how public and private workers are not separate tribes, but in the same households, parents, partners, sons and daughters. All use public services and many move fluidly between jobs in both sectors, truly all in this together.